CAT 2022 Slot 3 QA Question 8: If $3 + 2 \sqrt{2}$ is a root of the equation $a x^2 + b x + c = 0$, and $4 + 2 \sqrt{3}$ is a root of the equation $a y^2 + m y + n = 0$, where a, b, c, m and n are integers, then the value of $\dfra

Question

If $3 + 2 \sqrt{2}$ is a root of the equation $a x^2 + b x + c = 0$, and $4 + 2 \sqrt{3}$ is a root of the equation $a y^2 + m y + n = 0$, where a, b, c, m and n are integers, then the value of $\dfra

Accepted Answer

Since the coefficients are integers and one root is a surd, the conjugate surd is the other root.
For the first equation, the roots are $3 + 2\sqrt{2}$ and $3 - 2\sqrt{2}$.
$	ext{Sum} = 6 = \dfrac{-b}{a}$
$b = -6a$
$	ext{Product} = 1 = \dfrac{c}{a}$
$c = a$
For the second equation, the roots are $